Distance-Based Phylogenetic Methods Around a Polytomy

Abstract

Abstract. Distance-based phylogenetic algorithms attempt to solve the NP-hard least squares phylogeny problem by mapping an arbitrary dissimilarity map representing biological data to a tree metric. The set of all dissimilarity maps is a Euclidean space properly containing the space of all tree metrics as a polyhedral fan. Outputs of distancebased tree reconstruction algorithms such as UPGMA and Neighbor-Joining are points in the maximal cones in the fan. Tree metrics with polytomies lie at the intersections of maximal cones.A phylogenetic algorithm divides the space of all dissimilarity maps into regions based upon which combinatorial tree is reconstructed by the algorithm. Comparison of phylogenetic methods can be done by comparing the geometry of these regions. We use polyhedral geometry to compare the local nature of the subdivisions induced by least squares phylogeny, UPGMA, and Neighbor-Joining. Our results suggest that in some circumstances, UPGMA and Neighbor-Joining poorly match least squares phylogeny when the true tree has a polytomy.

Reconstructing the Phylogeny

“…Eickmeyer et al [106] developed the geometric view of the BME and NJ methods and defined BME polytopes in order to compare the accuracy of the NJ method with the BME method. Similarly, Davidson and Sullivant [107] used the polyhedral subdivisions to compare the accuracy of the NJ, UPGMA (Unweighted Pair-Group Method with Arithmetic Mean) and Least Squares methods. Here we propose to extend this approach to cophylogenies, namely to joint BME polytopes.…”

L-Infinity Optimization to Linear Spaces and Phylogenetic Trees

“…Example 3.3. Figure 5 depicts three ultrametrics in the set of closest ultrametrics to δ = (2,4,6,8,10,12). The subdominant ultrametric is δ U = (2, 4, 6, 4, 6, 6), d(δ, U 4 ) = 3, and the canonical closest ultrametric (pictured far left) is δ c = (5,7,9,7,9,9).…”

“…The idea behind most of these proofs is to find a linear space that contains the ultrametrics for many different tree topologies and apply the constructions for linear spaces developed in Section 2. In [6], the authors study the geometry of the set of dissimilarity maps around a polytomy with respect to the Euclidean norm. They showed that locally this space could be partitioned according to the closest tree topology.…”

“…We will perturb u to construct a dissimilarity map δ where the set of l ∞ -closest points to δ has dimension two. The triple (C(AB)) is compatible with T (u) and so we let δ = (5,7,9,7,9,9) + (e AC − e BC ) = (5,8,9,6,9,9).…”

“…The subdominant ultrametric δ U = (5,6,9,6,9,9) and the canonical ultrametric δ c = (6, 7, 10, 7, 10, 10). We have two degrees of freedom that come from adjusting the values of {(δ c ) AD , (δ c ) BD , (δ c ) CD } (sliding down the root) or {(δ c ) AB } (sliding down the most recent common ancestor of A and B).…”